Do apteki jest dalej niż do sklepu.

Because the sentence is talking about distance to a place.

• do apteki = to the pharmacy
• do sklepu = to the shop/store

In Polish, do is the normal preposition when you mean movement or direction toward many enclosed places, buildings, or destinations. Here it does not mean actual movement is happening right now; it just marks the destination in a distance comparison.

They are in the genitive case.

That is because the preposition do requires the genitive:

• apteka → do apteki
• sklep → do sklepu

So a learner should think of do as a preposition that automatically changes the noun into genitive.

Because they belong to different noun patterns.

• apteka is a feminine noun ending in -a, and feminine nouns like this often form the genitive singular in -y or -i:
  apteka → apteki
• sklep is a masculine inanimate noun, and many nouns of this type form the genitive singular in -u:
  sklep → sklepu

So the case is the same, but the ending depends on the noun’s gender and declension pattern.

Polish often leaves out the kind of dummy subject English uses.

In English, you say:

• It is farther to the pharmacy than to the shop.

In Polish, you can simply say:

• Do apteki jest dalej niż do sklepu.

The idea of it is understood from context, so Polish does not need to say it. This is very common.

Here dalej means farther/further away.

It is the comparative form related to daleko (far). So:

• daleko = far
• dalej = farther / further

In this sentence, jest dalej means is farther away.

Because niż means than after a comparative.

Since dalej is comparative (farther), Polish uses niż to introduce the second part of the comparison:

• dalej niż = farther than

So:

• Do apteki jest dalej niż do sklepu
  = It’s farther to the pharmacy than to the shop.

The starting point is understood from context.

The sentence really means something like:

• From here, it’s farther to the pharmacy than to the shop.
• From our house, it’s farther to the pharmacy than to the shop.

Polish often leaves that part out if it is obvious. If you want to make it explicit, you can say:

• Stąd do apteki jest dalej niż do sklepu.
  = From here, it’s farther to the pharmacy than to the shop.

Yes. Polish word order is more flexible than English word order.

The given sentence:

• Do apteki jest dalej niż do sklepu

is natural and neutral.

You may also hear other orders, depending on emphasis, but the version you have is a very good standard model. In Polish, changing the order often changes what is emphasized, not the basic meaning.

You could, but it is not exactly the same structure.

• Apteka jest dalej niż sklep compares the pharmacy and the shop more directly as places.
• Do apteki jest dalej niż do sklepu emphasizes the distance to each place from some understood starting point.

The do ... niż do ... version is clearer when you specifically want to talk about how far you have to go.

Because the sentence is about distance to a destination, not location in a place.

• do apteki = to the pharmacy
• w aptece = in the pharmacy
• na aptekę is not the normal expression here

So if you are talking about how far away a place is as a destination, do is the correct preposition.

Yes, it is natural.

A Polish speaker could absolutely say:

• Do apteki jest dalej niż do sklepu.

If needed, they might add the starting point:

• Stąd do apteki jest dalej niż do sklepu.
• Z domu do apteki jest dalej niż do sklepu.

But your original sentence is grammatically normal and sounds natural when the context makes the starting point clear.